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Mirrors > Home > ILE Home > Th. List > suplocexprlemell | GIF version |
Description: Lemma for suplocexpr 7533. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Ref | Expression |
---|---|
suplocexprlemell | ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 6055 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5347 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 1st Fn V |
4 | ssv 3119 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | fnssres 5236 | . . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴) | |
6 | 3, 4, 5 | mp2an 422 | . . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴 |
7 | eluniimadm 5666 | . . 3 ⊢ ((1st ↾ 𝐴) Fn 𝐴 → (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥))) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥)) |
9 | resima 4852 | . . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴) | |
10 | 9 | unieqi 3746 | . . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴) |
11 | 10 | eleq2i 2206 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴)) |
12 | fvres 5445 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥)) | |
13 | 12 | eleq2d 2209 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥))) |
14 | 13 | rexbiia 2450 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
15 | 8, 11, 14 | 3bitr3i 209 | 1 ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ∃wrex 2417 Vcvv 2686 ⊆ wss 3071 ∪ cuni 3736 ↾ cres 4541 “ cima 4542 Fn wfn 5118 –onto→wfo 5121 ‘cfv 5123 1st c1st 6036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-1st 6038 |
This theorem is referenced by: suplocexprlemml 7524 suplocexprlemrl 7525 suplocexprlemdisj 7528 suplocexprlemloc 7529 suplocexprlemex 7530 suplocexprlemlub 7532 |
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